Okay - how weirdly cool is this? There's a special DOFF mission where you have to actually go down to the engineering deck in person to talk to the operations officer.
The mission? Put on a holoplay of Hamlet. You basically have to pick and choose from 19 of your DOFF officers and pick out officers who can put on a play well enough to be successful given the necessary traits. I thought it was wacky enough to give it a try.

I aced it on my first go-through. Critical Success. And this was my reward:

Yes - that's right - a holographic WILLIAM SHAKESPEARE is now part of my crew!

How freaking cool is that? ^_^

I'm going to try again and see what else I might get. Wonder if I can get an entire cast of Holo-players eventually? ^_^

Okay, somebody please explain to me how my attempt to put on a production of Hamlet resulted in one death and four injuries? What the friggin' hell happened on that set? Did they use real poison or something?

And goddammit, that's the first duty officer I've ever lost. In a goddamned play.
-- Bob
---------
Then the horns kicked in...
...and my shoes began to squeak.

Remember, your hand at the level of your eyes!
--
"Anko, what you do in your free time is your own choice. Use it wisely. And if you do not use it wisely, make sure you thoroughly enjoy whatever unwise thing you are doing." - HymnOfRagnorok as Orochimaru at SpaceBattleswoot Med. Eng., verb, 1st & 3rd pers. prsnt. sg. know, knows

Okay, so basically there's a chance you can lose grey common or green uncommon duty officers.
Blue rare and purple very rare officers are effectively immortal; they will simply be injured and sent to sickbay to wait out the regular timer. You won't lose them. But you can lose the greyshirts and greenshirts. It doesn't happen often, but it can happen on a mission where there's a chance - however low - of injury.
-- Acyl

Yes - if you want to "trade up" it's a 5 to 1 progression all the way.
5 Grey = 1 Green5 Green = 1 Blue5 Blue = 1 Purple
Grey DOFFs used to be pretty cheap to get on the market, but now that they're used as fodder for the Starbase system they're more expensive.
The best way to get DOFFS if you're on the cheap is to go to the various recuitment officers at either the Starfleet or KDF academies. I'm not sure if I recall where you do that for Rom Captains. I think it's in the New Romulus area (the exclusively Romulan section).
Once there, look around and find a Personnel Officer. You can either do that 5-1 trade-off mission there or do a recruitment mission. The recruitment missions will tend to get you a lot of grey officers. And you have to have some patience with it. But hey - it's free.

Oh, yeah, I've been doing that already. Also my fleet starbase has a personnel officer who doesn't know about the other one and offers me the same recruitment options. So I double-dip as often as I can.
-- Bob
---------
Then the horns kicked in...
...and my shoes began to squeak.

Oh - my KDF Captain also had a failure with this mission too. But nothing spectacular happened. No deaths, no injuries. Just "substandard performance".
Or maybe it was substandard BECAUSE there were no deaths or injuries. This IS a Klingon ship after all... ^_^